Genus and arithmetic genus of a curve

Assume the Axiom of Choice for the coherence and cohomology routes below (The Axiom of Choice); it supplies Dependent Choice where the proper cohomology-finiteness route requires it (AC implies DC implies countable choice).

Let $k$ be a field and let $C$ be a smooth proper geometrically connected curve over $k$ (Curves over a field). Its genus is $g(C):=h^1(C,{\mathcal{O}}_C)={\dim }_k{H}^1(C,{\mathcal{O}}_C),$ the $k$-dimension of the degree-one sheaf cohomology of the structure sheaf (Sheaf cohomology as right derived global sections, Finite-dimensional vector space, and its dimension ${\dim }_{F}V$; infinite-dimensional means having no finite basis). The structure sheaf is coherent: it is quasi-coherent of finite type on the locally Noetherian scheme $C$ (Quasi-coherent module on a scheme, Finite type and finitely presented module sheaves, Locally Noetherian and Noetherian schemes, Coherent sheaves on a locally Noetherian scheme). Since $C$ is proper, coherent cohomology is finite-dimensional over $k$ (Proper morphisms, Finite-dimensional coherent cohomology over a field). By Functions on a proper curve one has $H^0(C,{\mathcal{O}}_C)=k$, so $g(C)=1-\chi ({\mathcal{O}}_C),$ where $\chi ({\mathcal{O}}_C)=h^0(C,{\mathcal{O}}_C)-h^1(C,{\mathcal{O}}_C)$ is the Euler characteristic of the coherent sheaf ${\mathcal{O}}_C$ (Euler characteristic of a coherent sheaf, Coherent module sheaves). Thus $g(C)$ is a finite integer.

For any integral proper finite-type $k$-scheme $X$ whose underlying Noetherian topological space has dimension one, define the arithmetic genus $p_a(X):=1-\chi ({\mathcal{O}}_X)=h^1(X,{\mathcal{O}}_X)-h^0(X,{\mathcal{O}}_X)+1.$ The structure sheaf ${\mathcal{O}}_X$ is coherent: $X$ is locally Noetherian because a field is Noetherian and finite-type algebras over it are Noetherian; it is quasi-compact because it is proper, and ${\mathcal{O}}_X$ is quasi-coherent of finite type (Locally Noetherian and Noetherian schemes, A field has only the zero ideal and itself, hence is Noetherian, Every algebra of finite type over a Noetherian ring is a Noetherian ring, Proper morphisms, Quasi-coherent module on a scheme, Finite type and finitely presented module sheaves, Coherent sheaves on a locally Noetherian scheme). Proper cohomology finiteness makes $H^0(X,{\mathcal{O}}_X)$ and $H^1(X,{\mathcal{O}}_X)$ finite- dimensional over $k$ (Finite-dimensional coherent cohomology over a field). The Noetherian topological-space dimension theorem gives $H^q(X,{\mathcal{O}}_X)=0$ for every $q\ge 2$, since $\dim X=1$ (Noetherian topological spaces via ACC on opens or DCC on closed subsets, Chain dimension and the empty-space convention, Grothendieck vanishing on a Noetherian space). Hence $\chi ({\mathcal{O}}_X)$ is the finite integer $h^0(X,{\mathcal{O}}_X)-h^1(X,{\mathcal{O}}_X)$, and so is $p_a(X)$ (Euler characteristic of a coherent sheaf). This definition requires only integrality, properness, finite type, and dimension one; $X$ need not be smooth or geometrically integral.

For a smooth proper geometrically connected curve $C$, the two invariants agree, $p_a(C)=g(C)$, because $H^0(C,{\mathcal{O}}_C)=k$ by Functions on a proper curve. The arithmetic genus is defined for singular integral proper curves as well, while $g(C)$ is the smoothness-dependent invariant.